Take a time series \((X_n)_{n \in \mathbb {Z}}\)  which is weakly stationary: its autocorrlelation function \(\gamma (k) := \mathbb {E}[X_0X_k] = \mathbb {E}[X_tX_{t+k}]\)  does not depend on \(t \in \mathbb {Z}\) . There exist a rich theory of these models with several classes of models: autoregressions, moving average models, their generalizations ARIMA (autoregressive integrated moving averages), stochastic volatility, generalized autoregressive conditional heteroscedastic GARCH, and others. We refer readers to classic textbooks: a comprehensive monograph [1]} and its more applied version by the same authors, [2]}; and a textbook [3]} with emphasis on finance and connections to continuous-time models. See also exposition of time series in [4]} with connection to linear regression models, economics, and finance.
